Dynamics
More specifically, the dynamics of the basic contact process is defined by the following transition rates: at site ,
where the sum is over all the neighbors in of . This means that each site waits an exponential time with the corresponding rate, and then flips (so 0 becomes 1 and viceversa).
For each graph there exists a critical value for the parameter so that if then the 1's survive (that is, if there is at least one 1 at time zero, then at any time there are ones) with positive probability, while if then the process dies out. For contact process on the integer lattice, a major breakthrough came in 1990 when Bezuidenhout and Grimmett showed that the contact process also dies out at the critical value. Their proof makes use of percolation theory.
Read more about this topic: Contact Process (mathematics)
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